Semi-smooth Newton methods for variational inequalities of the first kind
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 1, pp. 41-62.

Semi-smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.

DOI: 10.1051/m2an:2003021
Classification: 49J40,  65K10
Keywords: semi-smooth Newton methods, contact problems, variational inequalities, bilateral constraints, superlinear convergence
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     title = {Semi-smooth {Newton} methods for variational inequalities of the first kind},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
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     publisher = {EDP-Sciences},
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Ito, Kazufumi; Kunisch, Karl. Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 1, pp. 41-62. doi : 10.1051/m2an:2003021. http://www.numdam.org/articles/10.1051/m2an:2003021/

[1] D.P. Bertsekas, Constrained Optimization and Lagrange Mulitpliers. Academic Press, New York (1982). | MR

[2] M. Bergounioux, M. Haddou, M. Hintermüller and K. Kunisch, A comparison of a Moreau-Yosida based active strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11 (2000) 495-521. | Zbl

[3] M. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37 (1999) 1176-1194. | Zbl

[4] Z. Dostal, Box constrained quadratic programming with proportioning and projections. SIAM J. Optim. 7 (1997) 871-887. | Zbl

[5] R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer Verlag, New York (1984). | MR | Zbl

[6] R. Glowinski, J.L. Lions and T. Tremolieres, Analyse Numerique des Inequations Variationnelles. Vol. 1, Dunod, Paris (1976). | Zbl

[7] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as semi-smooth Newton method. SIAM J. Optim. (to appear). | Zbl

[8] R. Hoppe, Multigrid algorithms for variational inequalities. SIAM J. Numer. Anal. 24 (1987) 1046-1065. | Zbl

[9] R. Hoppe and R. Kornhuber, Adaptive multigrid methods for obstacle problems. SIAM J. Numer. Anal. 31 (1994) 301-323. | Zbl

[10] K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth convex optimization in Hilbert spaces. Nonlinear Anal. 41 (2000) 573-589. | Zbl

[11] K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343-364. | Zbl

[12] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). | MR | Zbl

[13] D.M. Troianello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987). | Zbl

[14] M. Ulbrich, Semi-smooth Newton methods for operator equations in function space. SIAM J. Optim. (to appear). | Zbl

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